[erlang-questions] Gaussian Distribution
Mon Oct 29 02:52:28 CET 2012
On 29/10/2012, at 4:58 AM, Frank Recker wrote:
> at work, I often need the values the cumulative distribution function of
> the Gaussian distribution. The code for this function in haskell, erlang
> and perl and the corresponding mathematical paper can be found at
> git://github.com/frecker/gaussian-distribution.git .
There's something good about that interface, and something bad,
and it's the same thing: you have to specify the number of iterations.
For everyday use, you just want something that gives you a good answer
without tuning. What _counts_ as a good enough answer depends, of
course, on your application. I adapted John D. Cook's C++ code and
used R-compatible names. (What I _really_ wanted this for was
Smalltalk. The Erlang code is new.) Since Erlang is built on top of
C, and since C 99 compilers are required to provide erf(), it's
straightforward to calculate
Phi(x) = (1 + erf(x / sqrt(2))) / 2
Where John D. Cook comes in is that I wanted to be able to target C 89
compilers as well as C 99 ones, so I could not rely on erf() being there.
Experimentally, the absolute error of pnorm/1 is below 1.0e-7 over the
range -8 to +8.
dnorm/1, % Density of Normal(0, 1) distribution at X
dnorm/3, % Density of Normal(M, S) distribution at X
erf/1, % The usual error function
pnorm/1, % Cumulative probability of Normal(0, 1) from -oo to X
pnorm/3 % Cumulative probability of Normal(M, S) from -oo to X
0.39894228040143267794 * math:exp((X*X)/2.0).
dnorm(X, M, S) ->
% Phi(x) = (1+erf(x/sqrt 2))/2.
% The absolute error is less than 1.0e-7.
(erf(X * 0.70710678118654752440) + 1.0) * 0.5.
pnorm(X, M, S) ->
% The following code was written by John D. Cook.
% The original can be found at http://www.johndcook.com/cpp_erf.html
% It is based on formula 7.1.26 of Abramowitz & Stegun.
% The absolute error seems to be less than 1.4e-7;
% the relative error is good except near 0.
if X < 0 ->
S = -1.0, A = -X
; true ->
S = 1.0, A = X
T = 1.0/(1.0 + 0.3275911*A),
Y = 1.0 - (((((1.061405429*T - 1.453152027)*T) + 1.421413741)*T -
0.284496736)*T + 0.254829592)*T*math:exp(-A*A),
S * Y.
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